Abstract
There exist two main versions of preconditioners of algebraic multilevel type, the additive and the multiplicative methods. They correspond to preconditioners in block diagonal and block matrix factorized form, respectively. Both can be defined and analysed as recursive two-by-two block methods. Although the analytical framework for such methods is simple, for many finite element approximations it still permits the derivation of the strongest results, such as optimal, or nearly optimal, rate of convergence and optimal, or nearly optimal order of computational complexity, when proper recursive global orderings of node points have been used or when they are applied for hierarchical basis function finite element methods for elliptic self-adjoint equations and stabilized in a certain way. This holds for general elliptic problems of second order, independent of the regularity of the problem, including independence of discontinuities of coefficients between elements and of anisotropy. Important ingredients in the methods are a proper balance of the size of the coarse mesh to the finest mesh and a proper solver on the coarse mesh. This paper presents in a survey form the basic results of such methods and considers in particular additive methods. This method has excellent parallelization properties.
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References
O. Axelsson, On multigrid methods of the two-level type, in: Multigrid Methods, Proceedings, Köln-Porz (1981), eds. W. Hackbusch and U. Trottenberg, Lecture Notes in Mathematics, Vol. 960 (Springer, Berlin, 1982) pp. 352–367.
O. Axelsson, An algebraic framework for hierarchical basis function multilevel methods or the search for ‘optimal’ preconditioners, in: Iterative Methods for Large Linear Systems, eds. D.R. Kincaid and L.J. Hayes (Academic Press, New York, 1990) pp. 7–40.
O. Axelsson, The stabilized V-cycle method, J. Comput. Appl. Math. 74 (1996) 33–40.
O. Axelsson, Stabilization of algebraic multilevel iteration methods; additive methods, in: AMLI '96, Proc. of the Conf. on Algebraic Multilevel Iteration Methods with Applications, eds. O. Axelsson and B. Polman, University of Nijmegen, The Netherlands (June 13–15, 1996) pp. 49–62.
O. Axelsson and V.A. Barker, Finite Element Solution of Boundary Value Problems. Theory and Computation (Academic Press, Orlando, FL, 1984).
O. Axelsson and V. Eijkhout, The nested recursive two-level factorization method for nine-point difference matrices, SIAM J. Sci. Statist. Comput. 12 (1991) 1373–1400.
O. Axelsson and I. Gustafsson, Preconditioning and two-level multigrid methods of arbitrary degree of approximation, Report 8120 (July 1981), Department of Mathematics, University of Nijmegen, The Netherlands. Later published in Math. Comp. 40 (1983) 219–242.
O. Axelsson and M. Neytcheva, Algebraic multilevel iteration method for Stieltjes matrices, Numer. Linear Algebra Appl. 1 (1994) 213–236.
O. Axelsson and M. Neytcheva, Scalable algorithms for the solution of Navier's equations of elasticity, J. Comput. Appl. Math. 63 (1995) 149–178.
O. Axelsson, M. Neytcheva and B. Polman, The bordering method as a preconditioning method, Vestnik Moscov. Univ. Ser. 15 Vichisl. Mat. Kybernet. (1995) 3–24.
O. Axelsson and A. Padiy, On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems, SIAM J. Sci. Comput., to appear.
O. Axelsson and P. Vassilevski, A survey of multilevel preconditioned iterative methods, BIT 29 (1989) 769–793.
O. Axelsson and P. Vassilevski, Algebraic multilevel preconditioning methods I, Numer. Math. 56 (1989) 157–177.
O. Axelsson and P. Vassilevski, Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590.
O. Axelsson and P. Vassilevski, Asymptotic work estimates for AMLI methods, Appl. Numer. Math. 7 (1991) 437–451.
O. Axelsson and P.S. Vassilevski, Variable step multilevel preconditioning methods, I: Self-adjoint and positive definite elliptic problems, Numer. Linear Algebra Appl. 1 (1994) 75–101.
Z. Bai and O. Axelsson, A unified framework for the construction of various algebraic multilevel preconditioning methods, in: AMLI '96, Proc. of the Conf. on Algebraic Multilevel Iteration Methods with Applications, eds. O. Axelsson and B. Polman, University of Nijmegen, The Netherlands (June 13–15, 1996) pp. 63–76.
N.S. Bakhvalov, On the convergence of a relaxation method with natural constraints on the elliptic operator, Comput. Math. Math. Phys. 6 (1966) 101–135.
R. Bank and T. Dupont, Analysis of a two-level scheme for solving finite element equations, Report (NA-159), Center for Numerical Analysis, The University of Texas at Austin (1980).
D. Braess, The contraction number of a multigrid method for solving the Poisson equation, Numer. Math. 17 (1981) 387–404.
J.H. Bramble, Multigrid Methods, Pitman Research Notes in Mathematics Series, Vol. 294 (Longman, New York, 1993).
A. Brandt, Multi-level adaptive solution to boundary-value-problems, Math. Comp. 31 (1977) 333–390.
R.P. Fedorenko, A relaxation method for solving elliptic difference equations, Comput. Math. Math. Phys. 1 (1962) 1092–1096.
W. Hackbusch, Multigrid convergence theory, in: Multigrid Methods, Proceedings, Köln-Porz (1981), eds. W. Hackbusch and U. Trottenberg, Lecture Notes in Mathematics, Vol. 960 (Springer, Berlin, 1982) pp. 177–219.
W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations (Springer, New York, 1994).
Y. Kuznetsov, Multigrid domain decomposition methods for elliptic problems, Comput. Methods Appl. Mech. Engrg. 75 (1989) 185–193.
J.F. Maitre and F. Musy, The contraction number of a class of two-level methods; An exact evaluation for some finite element subspaces and model problems, in: Multigrid Methods, Proceedings, Köln-Porz (1981), eds. W. Hackbusch and U. Trottenberg, Lecture Notes in Mathematics, Vol. 960 (Springer, Berlin, 1982) pp. 535–544.
P. Vassilevski, Multilevel preconditioning matrices and multigrid V-cycle methods, in: Proc. of the 4th GAMM-Seminar, Kiel (January 22–24, 1988), ed. W. Hackbusch, Notes on Numerical Fluid Mechanics, Vol. 23 (Vieweg, Braunschweig, 1988) pp. 200–208.
P. Vassilevski, Hybrid V-cycle algebraic multilevel preconditioners, Math. Comp. 58 (1992) 489–512.
H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986) 379–412.
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Axelsson, O. Stabilization of algebraic multilevel iteration methods; additive methods. Numerical Algorithms 21, 23–47 (1999). https://doi.org/10.1023/A:1019136808500
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DOI: https://doi.org/10.1023/A:1019136808500